ABACABA pattern

In order to generate the next sequence, first take the previous pattern, add the next letter from the alphabet, and then repeat the previous pattern. The first few steps are listed here.

ABACABA is a "quickly growing word", often described as chiastic or "symmetrically organized around a central axis" (see: Chiastic structure and Χ). The number of members in each iteration is {{math|1=a(n) = 2ⁿ − 1}}, the Mersenne numbers ({{oeis|A000225}}).

{{Gallery

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| File:ABACABA Sierpinski triangle.svg

| alt1=

| Sierpinski triangle:
{{center|ABACABA}}

| File:Measuring - Fractions of an inch.svg

| alt2=

| Ruler, excluding 1 and 2:
{{center|ABACABADABACABA}}excluding 2:
{{center|EABACABADABACABA}}

| File:Cantor4 abacaba pattern.png

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| Cantor set:
{{center|ABACABADABACABA}}

| File:Ex 001.png

| alt4=

| Binary tree/upside down family tree:
{{center|ABACABADABACABA}}

| File:KochCurve1 abacaba pattern.png

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| Koch curve: $n=1$ is ABA, $n=2$ is ABACABA, and $n=3$: ABACABADABACABA

| File:Solfege subdivision de la ronde a la croche.svg

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| Metric hierarchy:
{{center|ABACABADABACABA{{efn|The strength, emphasis, or importance of the beginning of each duration $1/8$ the length of a single measure in common time (eighth-notes) is, divisively ($2/2^1=1$, $4/2^2=1$, $8/2^3=1$), determined by each eighth-note's position in a DABACABA structure, while the eighth notes of two measures grouped, additively ($8\backslash times2=16$), are determined by an EABACABADABACABA structure.}}}}

| File:Metric levels.svg

| alt7=

| Metric levels: {{center|EABACABADABACABA}}

| File:abacaba_binary.svg

| alt8=

| When counting in binary (here 4-bit), the final 0s form an ABACABA pattern

| File:Staircase of largest squares abacaba pattern.svg

| alt9=

| A staircase built with the largest possible squares/cubes while allowing equally sized steps: {{center|ABACABADABACABA}}

| File:Circle-box fractal abacaba pattern.svg

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| A "circle fractal" superimposed with a {{math|2 × 2}} box fractal: {{center|ABACABADABACABA}}

| File:Tower of Hanoi recursion SMIL.svg

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| The Tower of Hanoi with four disks: {{center|ABACABADABACABA}}

| File:Binary tree as array.svg

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| Binary tree array: {{center|to O}}

| File:BRGC Mask.svg

| alt13=

| Binary-reflected Gray code (BRGC): {{center|to G}}

| File:DMT Winkelkod-abs.svg

| alt14=

| Rotary encoder: {{center|to I}}

| File:Blue cube with Gray code path.png

| alt15=

| 3-bit Gray code visualized as a traversal of vertices of a cube (0,1,3,2,6,7,5,4): {{center|ABACABA}}

| File:Double harmonic scale marked.png

| alt16=

| Double harmonic scale ({{audio|Double harmonic scale.mid|Play}}) with steps of {{nobreak|H-3H-H-W-H-3H-H}}: {{center|ABACABA}}

| File:Chambord Castle Northwest facade.jpg

|alt17=

| Château de Chambord: {{center|ABACABAWright, Craig (2016). Listening to Western Music, p.48. Cengage Learning. {{ISBN|9781305887039}}.}}

| File:Gray code number line arcs.svg

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| Gray code along the number line ({{oeis|A003188}}): {{center|ABACABADABACABAEABACABADABACABA}}

| File:Baguenaudier.svg

| alt19=

| Devil's needle: {{center|ABACABADABACABA}}

| File:Menger_sponge_diagonal_section.png

| alt20=

| Size of hexagrams on a diagonal of a section of a Menger sponge model: {{center|ABACABADABACABA}}

}}

• Arch form
• Farey sequence
• Rondo
• Sesquipower

{{notelist}}

{{Reflist}}

• Naylor, Mike: [http://abacaba.org/ abacaba.org]

Category:Fractals

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